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arXiv:cond-mat/0601432 · Physical Review A

Atomic matter of nonzero-momentum Bose-Einstein condensation and orbital current order

W. Vincent Liu, Congjun Wu

Published Jul 13, 2006• 155 citations
arXivHTMLPDF

Abstract

The paradigm of Bose-Einstein condensation has been associated with zero momentum to which a macroscopic fraction of bosons condense. Here we propose a new quantum state where bosonic alkali-metal atoms condense at nonzero momenta, defying the paradigm. This becomes possible when the atoms are confined in the $p$-orbital Bloch band of an optical lattice rather than the usual $s$-orbital band. The new condensate simultaneously forms an order of transversely staggered orbital currents, reminiscent of orbital antiferromagnetism or $d$-density wave in correlated electronic systems but different in fundamental ways. We discuss several approaches of preparing atoms to the $p$-orbital band and propose an ``energy blocking'' mechanism by Feshbach resonance to protect them from decaying to the lowest $s$-orbital band. Such a model system seems very unique and novel to atomic gases. It suggests a new concept of quantum collective phenomena of no prior example from solid state materials.

DOI
10.1103/physreva.74.013607

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The paradigm of Bose-Einstein condensation has been associated with zero momentum to which a macroscopic fraction of bosons condense. Here we propose a new quantum state where bosonic alkali-metal atoms condense at nonzero momenta, defying the paradigm. This becomes possible when the atoms are confined in the ppp-orbital Bloch band of an optical lattice rather than the usual sss-orbital band. The new condensate simultaneously forms an order of transversely staggered orbital currents, reminiscent of orbital antiferromagnetism or ddd-density wave in correlated electronic systems but different in fundamental ways. We discuss several approaches of preparing atoms to the ppp-orbital band and propose an ``energy blocking'' mechanism by Feshbach resonance to protect them from decaying to the lowest sss-orbital band. Such a model system seems very unique and novel to atomic gases. It suggests a new concept of quantum collective phenomena of no prior example from solid state materials.
Abstract This article provides a synopsis of our recent experimental work exploring Bose–Einstein condensation in metastable higher Bloch bands of optical lattices. Bipartite lattice geometries have allowed us to implement appropriate band structures, which meet three basic requirements: the existence of metastable excited states sufficiently protected from collisional band relaxation, a mechanism to excite the atoms initially prepared in the lowest band with moderate entropy increase, and the possibility of cross-dimensional tunneling dynamics, necessary to establish coherence along all lattice axes. A variety of bands can be selectively populated and a subsequent thermalization process leads to the formation of a condensate in the lowest energy state of the chosen band. As examples the 2nd, 4th and 7th bands in a bipartite square lattice are discussed. The geometry of the 2nd and 7th bands can be tuned such that two inequivalent energetically degenerate energy minima arise at the X ± -points at the edge of the 1st Brillouin zone. In this case even a small interaction energy is sufficient to lock the phase between the two condensation points such that a complex-valued chiral superfluid order parameter can emerge, which breaks time reversal symmetry. In the 4th band a condensate can be formed at the Γ-point in the center of the 1st Brillouin zone, which can be used to explore topologically protected band touching points. The new techniques to access orbital degrees of freedom in higher bands greatly extend the class of many-body scenarios that can be explored with bosons in optical lattices.
Feynman's "no-node" theorem states that the conventional many-body ground state wavefunctions of bosons in the coordinate representation are positive definite. This implies that time-reversal symmetry cannot be spontaneously broken. In this article, we review our progress in studying a class of new states of unconventional Bose–Einstein condensations beyond this paradigm. These states can either be the long-lived metastable states of ultracold bosons in high orbital bands in optical lattices as a result of the "orbital Hund's rule" interaction, or the ground states of spinful bosons with spin-orbit coupling linearly dependent on momentum. In both cases, Feynman's argument does not apply. The resultant many-body wavefunctions are complex-valued and thus break time-reversal symmetry spontaneously. Exotic phenomena in these states include the Bose–Einstein condensation at nonzero momentum, the ordering of orbital angular momentum moments, the half-quantum vortex, and the spin texture of skyrmions.
Understanding strongly correlated quantum materials, such as high-T_{c} superconductors, iron-based superconductors, and twisted bilayer graphene systems, remains as one of the outstanding challenges in condensed matter physics. Quantum simulation with ultracold atoms in particular optical lattices, which provide orbital degrees of freedom, is a powerful tool to contribute new insights to this endeavor. Here, we report the experimental realization of an unconventional Bose-Einstein condensate of ^{87}Rb atoms populating degenerate p orbitals in a triangular optical lattice, exhibiting remarkably long coherence times. Using time-of-flight spectroscopy, we observe that this state spontaneously breaks the rotational symmetry and its momentum spectrum agrees with the theoretically predicted coexistence of exotic stripe and loop-current orders. Like certain strongly correlated electronic systems with intertwined orders, such as high-T_{c} cuprate superconductors, twisted bilayer graphene, and the recently discovered chiral density-wave state in kagome superconductors AV_{3}Sb_{5} (A=K, Rb, Cs), the newly demonstrated quantum state, in spite of its markedly different energy scale and the bosonic quantum statistics, exhibits multiple symmetry breakings at ultralow temperatures. These findings hold the potential to enhance our comprehension of the fundamental physics governing these intricate quantum materials.
We study a model of interacting bosons that occupy the first excited ppp-band states of a two-dimensional optical lattice. In contrast to the much studied single band Bose-Hubbard Hamiltonian, this more complex model allows for nontrivial superfluid phases associated with condensation at nonzero momentum and staggered order of the orbital angular momentum in addition to the superfluid-Mott insulator transition. More specifically, we observe staggered orbital angular momentum order in the Mott phase at commensurate filling and superfluidity at all densities. We also observe a transition between the staggered angular momentum superfluid phase and a striped superfluid, with an alternation of the phase of the superfluid along one direction. The transition between these two phases was observed in a recent experiment, which is then qualitatively well described by our model.
Superfluidity and Bose-Einstein condensation (BEC) in ultracold atomic gases represent macroscopic quantum phenomena arising from collective particle behavior at near-zero temperatures. Experimental realizations of BEC in alkali gases have enabled the observation of quantized vortices, matter-wave interference, and nonclassical excitations. This article reviews theoretical frameworks including Gross-Pitaevskii and Bogoliubov models, experimental techniques such as evaporative cooling and optical trapping, and computational approaches for simulating condensate dynamics. Two graphical analyses illustrate the condensate density profile and vortex lattice formation. Understanding these phenomena has implications for quantum simulation, atom interferometry, and exploration of quantum hydrodynamics.
Abstract Topological superfluidity is an important concept in electronic materials as well as ultracold atomic gases1. However, although progress has been made by hybridizing superconductors with topological substrates, the search for a material—natural or artificial—that intrinsically exhibits topological superfluidity has been ongoing since the discovery of the superfluid 3He-A phase2. Here we report evidence for a globally chiral atomic superfluid, induced by interaction-driven time-reversal symmetry breaking in the second Bloch band of an optical lattice with hexagonal boron nitride geometry. This realizes a long-lived Bose–Einstein condensate of 87Rb atoms beyond present limits to orbitally featureless scenarios in the lowest Bloch band. Time-of-flight and band mapping measurements reveal that the local phases and orbital rotations of atoms are spontaneously ordered into a vortex array, showing evidence of the emergence of global angular momentum across the entire lattice. A phenomenological effective model is used to capture the dynamics of Bogoliubov quasi-particle excitations above the ground state, which are shown to exhibit a topological band structure. The observed bosonic phase is expected to exhibit phenomena that are conceptually distinct from, but related to, the quantum anomalous Hall effect3–7 in electronic condensed matter.